The minimality of determinantal varieties

The determinantal variety {\Sigma_{pq}} is defined to be the set of all {p\times q} real matrices with {p\geq q} whose ranks are strictly smaller than q. It is proved that {\Sigma_{pq}} is a minimal cone in {\mathbb{R}^{pq}} and all its strata are regular minimal submanifolds.

MINIMAL HYPERSURFACES ASYMPTOTIC TO SIMONS CONES

In this paper, we prove that, up to similarity, there are only two minimal hypersurfaces in $\mathbb{R}^{n+2}$ that are asymptotic to a Simons cone, i.e., the minimal cone over the minimal hypersurface $\sqrt{\frac{p}{n}}\mathbb{S}^{p}\times \sqrt{\frac{n-p}{n}}\mathbb{S}^{n-p}$ of $\mathbb{S}^{n+1}$.

The electroretinogram of the rhodopsin knockout mouse

The electroretinogram (ERG) of the rhodopsin knockout (rho−/−) mouse of Humphries et al. (1997) (Humphries et al., 1997) was studied for evidence of light-evoked rod activity and to describe the cone function. The rho−/− retina develops normal numbers of rod and cone nuclei, but the rods have no outer segments, and no rhodopsin is found by immunohistochemistry. The dark-adapted ERG threshold was elevated 4.7 log units above wild-type (WT) control mice, indicating that any residual rod responses were reduced >50,000-fold, consistent with a complete functional knockout. The dark-adapted rho−/− ERG had a cone waveform, and the spectral sensitivity peaked near 510 nm for both dark-adapted and light-adapted conditions, without evidence of a Purkinje shift. The light-adapted ERG b-wave amplitude of young rho−/− mice was the same as WT. The amplitude remained steady up to postnatal day P47, but thereafter it declined to only 1–2% by P80 when no cone outer segments remained. Cone b-wave threshold of dark-adapted rho−/− mice was −1.07 ± 0.39 log cd-s/m2 (n = 17), which is 1.27 log units more sensitive than light-adapted thresholds against a rod-suppressing Ganzfeld background of 1.61 log scotopic cd/m2. This indicates that dark-adapted WT responses to still dimmer stimuli are exclusively rod driven with minimal cone intrusion. Above this cone threshold intensity, the dark-adapted b-wave of WT will be a summation of rod and cone responses. Threshold versus intensity (TVI) studies gave no evidence of a rod influence on the mouse cone b-wave.

Approximation by smooth embedded hypersurfaces with positive mean curvature

Here we initiate the study of the following problem. Let Ω be a compact domain in a Riemannian manifold such that ∂Ω is of minimum area for the contained volume. Can ∂Ω be approximated by smooth hypersurfaces of positive mean curvature? It reduces to the question of whether or not a stable (or minimizing) hypercone in a Euclidian space can be approximated by smooth hypersurfaces of positive mean curvature. The positive solution to the problem may be useful for studying the curvature and topology of Ω.We show in this paper that such approximation is possible provided that the given minimal cone satisfies some additional hypothesis.